I haven't seen a correlation with moon phase on the actual day of hunting, but the research shows that it can tell you something about the rut. The 2nd full moon after the autumn equinox is called the rut moon. It is when the rut is supposed to kick off. Most of the time it is the beginning of November, but this year it is this weekend. From my experience, it is when the little bucks start running around, but the big boys will still hold out for a bit.
As far as day to day hunting, I hunted Buffalo County, WI last year the first week of November. This is the #1 area in the world for Booners. That week was kicked off with a full moon. I hunted 5 days straight in my stand from sun-up to sun-down and I did not see a deer between 10 and 2:30.